Let R be the region bounded by the graph of y = e^(2x-x^2) and the horizontal line y = 2. What is the volume of R revolved about y = 1?

I gave my answer as pi{[e^(2x-x^2)]^2 - [2-e^(2x-x^2)]^2 dx with limits from 0.446 to 1.554.

Am I right?

April 28th 2009, 06:53 AM

apcalculus

Quote:

Originally Posted by Kaitosan

Let R be the region bounded by the graph of y = e^(2x-x^2) and the horizontal line y = 2. What is the volume of R revolved about y = 1?

I gave my answer as pi{[e^(2x-x^2)]^2 - [2-e^(2x-x^2)]^2 dx with limits from 0.446 to 1.554.

Am I right?

Inner Radius: 1

Outer Radius

Use the Washer Method:

PI INTEGRAL (OUTER^2 - INNER^2) dx with the correct bounds, which you can find by intersecting the two curves. Your two bounds are correct.

Good luck!!

April 28th 2009, 07:30 AM

Kaitosan

I don't get it. It looks like you're including the area from y=e^(2x-x^2) to y=1, while in reality R is bounded by y=2 and y=e^(2x-x^2). Or maybe the problem allows for you to do both ways, since it looks like both ways satisfies the condition of being revolved around y=1.

April 28th 2009, 10:15 AM

apcalculus

Quote:

Originally Posted by Kaitosan

I don't get it. It looks like you're including the area from y=e^(2x-x^2) to y=1, while in reality R is bounded by y=2 and y=e^(2x-x^2). Or maybe the problem allows for you to do both ways, since it looks like both ways satisfies the condition of being revolved around y=1.

No I am not. including it. The distance from the rotational axis to the nearest boundary on the region is 1 unit (the distance from y=1 to y=2) at every level x. Call this the inner radius r(x).

The distance from the rotational axis to the outer boundary of the region is the y-coordinate of your exponential function minus 1, due the the rotational axis being y=1, one unit above the horizontal axis. Call this R(x).

No I am not. including it. The distance from the rotational axis to the nearest boundary on the region is 1 unit (the distance from y=1 to y=2) at every level x. Call this the inner radius r(x).

The distance from the rotational axis to the outer boundary of the region is the y-coordinate of your exponential function minus 1, due the the rotational axis being y=1, one unit above the horizontal axis. Call this R(x).